From imaging to material identification: A generalized concept of topological sensitivity
Introduction
Elastic-wave identification of defects and heterogeneities embedded in semi-infinite or finite bodies is a problem of considerable interest in mechanics owing to its applications in material characterization, seismology, and medical diagnosis. The underlying inverse solutions can be derived from a variety of computational platforms that include e.g. (far-field) ray theory (Aki and Richards, 2002), finite-difference approximation of the wave equation (Sheriff and Geldart, 1995, Schroeder et al., 2002), and boundary integral formulations (Bonnet, 1995). In the context of 3D material characterization, these approaches carry a substantial computational cost associated with solving the forward problem. This precludes the use of global search techniques such as genetic algorithms which involve a large number of forward simulations. To mitigate the problem, gradient-based optimization algorithms have been proposed as a computationally tractable alternative to solving inverse scattering problems, especially when aided by the analytical gradient estimates (Plessix et al., 1998, Guzina and Bonnet, 2004). Unfortunately, the latter class of solutions necessitate a reliable preliminary information about the geometry and material characteristics of hidden defects for satisfactory performance.
Building on the results in shape optimization obtained for Laplace (Sokolowski and Zochowski, 1999, Garreau et al., 2001) and Helmholtz (Samet et al., 2004, Pommier and Samet, 2005) systems, Guzina and Bonnet (2004), Bonnet and Guzina (2004), and Gallego and Rus (2004) have recently established the method of topological sensitivity as a tool for preliminary, grid-based reconstruction of obstacles in the context of inverse elastic scattering that requires no prior information (or assumptions in the absence thereof) about the location and geometry of internal defects. In the approach the topological derivative, which quantifies the sensitivity of a given cost functional with respect to the nucleation of an infinitesimal obstacle in the reference (background) medium, is used as an effective obstacle indicator through an assembly of sampling points where it attains pronounced negative values. Typically, the formulas for topological sensitivity permit an explicit representation (e.g. in terms of the Green's function) that is responsible for the computational efficiency of this class of reconstruction techniques. Notwithstanding their usefulness, however, the foregoing analyses are limited in the sense that they are focused on the nucleation of impenetrable scatterers, and in particular cavities in 3D (Guzina and Bonnet, 2004, Bonnet and Guzina, 2004) and/or cracks in 2D (Gallego and Rus, 2004) elastodynamics.
In this study, the concept of topological sensitivity is generalized to permit the nucleation of dissimilar solid inclusions and thus allow for preliminary elastic-wave identification of subsurface defects of more general nature. On employing a boundary integral approach to derive the necessary asymptotics in terms of the vanishing defect size, it is shown that the proposed generalization (termed “material-topological” sensitivity) consists of a monopole term, related to the mass density contrast, and a dipole term involving the elasticity contrast between the defect and the matrix. To cater for engineering applications, explicit formulas are derived for canonical cases when the nucleating inclusion takes spherical or ellipsoidal shape. For generality, the proposed developments are recast within an alternative framework of the adjoint-field formulation that permits nucleation of inclusions in an arbitrary (infinite or finite, homogeneous or heterogeneous) reference solid. Through numerical examples it is shown that the material-topological sensitivity can be used, in the context of inverse scattering, as an effective defect indicator through an assembly of sampling points where it attains marked negative values. On varying the material characteristics of the nucleating obstacle, it is also shown that the featured sensitivity can be used as a preparatory tool for both geometric and material identification of internal defects.
Beyond their intrinsic potential for the study of localized damage evolution in natural and engineered materials (e.g. Mazars et al., 1991, Xu, 2004, Bonamy et al., 2005), the proposed developments be especially useful in breast cancer detection wherein the knowledge of the shear modulus of a lesion, (geometrically) identified via ultrasound or magnetic resonance imaging, may permit reliable differentiation between the malignant and benign growths (Sarvazyan et al., 1998, Fatemi and Greenleaf, 1998). For completeness, it is noted that the underlying idea of a nucleating inclusion, explored in this study, is similar in spirit to the recent work in Ammari and Kang (2004) dealing with Laplace, elastostatic and Helmholtz systems. Notwithstanding the apparent commonalities, however, the asymptotic expansion methodology, formulas for the generalized topological sensitivity, and the material-geometric identification approach proposed herein have not been established elsewhere.
Section snippets
Preliminaries
Consider the inverse scattering problem where the semi-infinite solid, probed by elastic waves, contains a bonded defect with smooth boundary (see Fig. 1). With the Cartesian frame set at the top surface , the reference elastic half-space with closure is characterized by the shear modulus , Poisson's ratio , and mass density . Elastic parameters of the obstacle, , are and ; its mass density is denoted as .
Topological sensitivity
To aid the gradient-based minimization of (1) that is often used as a tool for identifying on the basis of motion measurements , of interest in this study is the development of material-topological derivative for the class of cost functionals given by (1) that would provide a reliable preliminary information about the location, geometry and material characteristics of the hidden defect. In situations when is a non-convex function in the material-geometric parametric space
Asymptotic for a nucleating inclusion
To arrive at a compact expression for (17) when is non-trivial, integral representation (12) when and can be conveniently rewritten by virtue of (10) aswhere denotes the unit normal on oriented toward the interior of ; , and , , and denote, respectively, the three integrals on the right-hand side of (22). Note that the order of multiplicands
Direct formulation
By virtue of (10) and (49), one may expand (1) with respect to the scattered field caused by the vanishing obstacle, , aswhereSince is by definition independent of a, one finds from (15), (49) and (50) that for the 3D nucleating inclusion problem; a behavior that conforms with the hypothesis made in (16). Although the choice of multiplicative constant in
Results and discussion
As shown in Guzina and Bonnet (2004) and Bonnet and Guzina (2004), the nucleating-cavity analogues of (54) and (74) can be used as a robust tool for the preliminary 3D reconstruction of impenetrable defects in the context of inverse scattering. Owing to their explicit dependence on the material properties of a nucleating inclusion, on the other hand, (54) and (74) carry an additional potential for material identification trough a parametric variation of geared toward minimizing the value of
Conclusions
In this study the concept of topological derivative, that has its origins in elastostatics and structural shape optimization, is extended to permit preliminary, yet robust 3D elastic-wave identification of material defects. In a departure from earlier studies that revolve around the idea of cavity nucleation, the proposed approach postulates the creation of an (infinitesimal) elastic inclusion. On taking the limit of a boundary integral representation of the scattered field caused by an elastic
Acknowledgments
The support provided by the National Science Foundation through Award No. CMS-0324348 to B. Guzina and the University of Minnesota Supercomputing Institute during the course of this investigation is kindly acknowledged.
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